Single photon emission and recombination dynamics in self-assembled GaN/AlN quantum dots

III-nitride quantum dots (QDs) are a promising system actively studied for their ability to maintain single photon emission up to room temperature. Here, we report on the evolution of the emission properties of self-assembled GaN/AlN QDs for temperatures ranging from 5 to 300 K. We carefully track the photoluminescence of a single QD and measure an optimum single photon purity of g(2)(0) = 0.05 ± 0.02 at 5 K and 0.17 ± 0.08 at 300 K. We complement this study with temperature dependent time-resolved photoluminescence measurements (TRPL) performed on a QD ensemble to further investigate the exciton recombination dynamics of such polar zero-dimensional nanostructures. By comparing our results to past reports, we emphasize the complexity of recombination processes in this system. Instead of the more conventional mono-exponential decay typical of exciton recombination, TRPL transients display a bi-exponential feature with short- and long-lived components that persist in the low excitation regime. From the temperature insensitivity of the long-lived excitonic component, we first discard the interplay of dark-to-bright state refilling in the exciton recombination process. Besides, this temperature-invariance also highlights the absence of nonradiative exciton recombinations, a likely direct consequence of the strong carrier confinement observed in GaN/AlN QDs up to 300 K. Overall, our results support the viability of these dots as a potential single-photon source for quantum applications at room temperature.


S1. µ-PL AND TRPL OPTICAL CHARACTERIZATION METHODS
Microphotoluminescence (µ-PL) and second order auto-correlation (g (2) (τ )) measurements were performed under continuous-wave (cw) excitation using either a 266 nm For TRPL measurements, the laser was focused using a conventional plano-convex lens of 2.5 cm focal length to reach a spot size of ≃ 100 µm. In order to optimize the collection of the sample photoluminescence, we guided each laser beam using a 90:10 transmission/reflection beam sampler onto the sample, which was placed in a closed-cycle helium cryostat (Cryostation C2 from Montana Instruments, Inc.) to cool it down to 5 K. µ-PL mapping of the  surface was performed by mounting the microscope objective on a Physik Instrumente P-612.2 XY piezostage. µ-PL spectra were recorded with a Horiba Symphony II UV-enhanced charge-coupled device (CCD) coupled to a Horiba FHR 640 monochromator equipped with 1800 l·mm −1 and 150 l·mm −1 holographic gratings. Autocorrelation measurements were performed using a Hanbury-Brown and Twiss (HBT) interferometer with two PicoQuant PMA 175 photomultiplier tubes linked to a PicoHarp 300 time-correlated single photon counter (TCSPC). For TPRL measurements, a PHD-400 photodiode module was alternatively connected to one channel of the TCSPC to use the laser pulses as a trigger. We measured a time resolution of 220 ps for the autocorrelation setup. The bandpass for g (2) and TRPL measurements was selected using a SPEX 270M monochromator with a 2400 l·mm −1 grating and a slit opening of 1.6 mm. The collection efficiency of the HBT setup was estimated to (0.27 ± 0.10) %. Further details can be found in Tamariz et al. [1].

S2. PHOTON EXTRACTION EFFICIENCY
Besides the HBT setup collection efficiency reported above, we should ideally compute the photon extraction efficiency from our sample to relate the detection rate to the real QD emission rate. While an exact computation of this latter efficiency goes beyond the scope of this work given the complexity introduced by the mesa geometry, we can nonetheless give a lower bound value for the extraction efficiency η ext,min by approximating our system to a conventional planar geometry while only considering photons emitted toward the top AlN/air interface, i.e., the photon flux partly reflected at the AlN/silicon substrate interface is neglected. Within this framework, the QD is considered as an isotropic emitter and the optical losses are only assumed to occur at the AlN/air interface. Owing to its wide bandgap, the AlN layer is fully transparent in the whole domain of emission of the QDs. The transmission coefficient T under normal incidence, also known as the dielectric efficiency [2], obtained using the refractive index of AlN n AlN ≈ 2.13 [3] around 4.5 eV, is defined as: η ext,min depends on the solid angle leading to light extraction Ω c , which simply writes: where θ c = arcsin n −1 AlN is the critical angle leading to total internal reflection and Ω tot = 4π is the total solid angle. As a result, we get η ext,min ≈ T · Ω c /Ω tot , which leads to η ext,min = 5.1 %. In pratice, we expect the actual extraction efficiency to be significantly higher. Note also that given the much smaller refractive index value of wide bandgap semiconductors compared to their III-arsenide counterparts, the present η ext,min value is more than three times larger than that we would obtain for InAs/GaAs QDs. polarization orientation (Fig. S3). The two QD A biexciton (XX) lines are not resolved, while the two exciton lines (X 1,2 ) are fully separated. The lines L 1 to L 3 have not been assigned so far, but they show the same linear polarization orientation than XX and X 2 . X 1 is crosspolarized to all other lines of QD A . Apart from L 3 , similar patterns have been observed earlier on GaN/AlN QDs [1,4,5].
The comparison between the polarization-dependent photoluminescence measurements of different QDs performed at cryogenic temperature suggests that the polarization of exciton emission lines could be pinned by crystallographic axes. As an illustration, Fig. S4 shows the polar representation of the normalized intensity of three X 1 exciton emission lines. The data are well reproduced when using the relationship [6]: where a, b and θ 0 are fitting parameters. The linear polarization degree ρ = (I max − I min )/(I max + I min ), where I max and I min are the maximum and minimum X 1 line integrated µ−PL intensity, respectively, is systematically above 95 %. The polarization axis of the QD A -X 1 line is tilted by about 60°with respect to QD B,C -X 1 , hence recalling the in-plane hexagonal symmetry of wurtzite GaN. However, we do not have a statistically significant set of data to fully validate this preliminary result. A previous report on the polarized emission of single GaN/AlN QDs by Bardoux and co-workers [6] did not support the existence of such preferential polarization axes while no unequivocal conclusion arose from the results reported in the PhD thesis by Kindel [7].
At 5 K, XX and X 1 dominate the emission spectrum. The transition from dark to bright states requires the simultaneous absorption of phonons which match their energy difference [4]. Assuming a negligible dark-state splitting, the intensity of X 2 increases when the thermal splitting. As observed in Fig. S5, X 2 takes over X 1 in intensity around 40 K. 80 K and 100 K spectra stems from a grating change. Low (up to T = 80 K) and high temperature spectra were recorded with a 1800 l·mm −1 and a 150 l·mm −1 grating, respectively.

S4. LIMITATIONS OF THE SECOND ORDER AUTO-CORRELATION FITTING FUNCTION AND IMPACT OF CHARGE FLUCTUATIONS
The standard multi-level model (Eq. 1 of the main text) used to fit the second order autocorrelation function measurements predicts a narrowing of the antibunching dip as a consequence of a bunching behavior. This arises from an increase in the QD mean occupation number µ = Π/γ, where the decay rate γ = 1/τ decay , with τ decay the exciton decay time, is assumed fixed and the pump rate Π depends on the excitation power density. This narrowing is visible in Fig. 2c of the main text. However, the fit partially diverges from the data when γ is kept constant. A more consistent fit can only be obtained when assuming an increase in γ with excitation power density. Such a fit yields a τ decay value that drops from about 19 to 13 ns between 40 and 400 kW cm −2 . This variation may be accounted for by different phenomena that add to the contribution of dark states to the exciton dynamics discussed in the main text. These alternative explanations are discussed hereafter.
In polar III-nitride QDs, the strong internal electric field E in due to the macroscopic polarization mismatch between the dot and the barrier materials is responsible for the out-of- where µ is the exciton dipole moment, which is aligned along the c-axis and is oriented toward the top of the QD in metal-polar III-nitride QDs. DSs can be modelled as traps with given capture τ ↓ and escape τ ↑ times. At cryogenic temperature and under low excitation conditions, the phonon-mediated escape of trapped carriers is blocked and DSs are assumed to be mostly charged [9,10]. Carrier escape is triggered when the sample is pumped and the higher the excitation power density, the lower the probability for a defect to be occupied. In the extreme situation where all the defects are depleted, the exciton energy gets ultimately shifted by an energy ∆E max , which accounts for the contribution of all DSs. As developed in Ref. [7], for DSs in the close vicinity of polar III-nitride QDs, the exciton peaks are expected to shift with increasing excitation power density following In individually. At first sight, such a decrease in τ decay would be commensurate with a noticeable change in the oscillator strength of the exciton ground state and hence a large spectral shift.
Indeed, in the energy range associated to these dots -QD A,C,D all emit above 4 eV-the energy shift leading to a change of 30 % in τ decay would amount to hundreds of meV (cf. Fig. 11 of the main text). Thus, any screening of the built-in electric field can be excluded to explain the measured variation in the decay time. Since the amplitude of the electric field generated by DSs is on the order of a few MV m −1 [11], it cannot impact the built-in electric field of about 7-9 MV cm −1 [8,12] significantly enough to explain the measured reduction in the decay time.
Besides variations in the built-in electric field experienced by the ground state exciton, another reason for explaining the divergence between the fit with fixed γ and experimental g (2) (τ ) data can also be found in the original hypothesis used to develop the multi-level model. Indeed, to allow for a closed analytic form of the second order autocorrelation function in the framework of this model, the recombination rate of an N -exciton is assumed to be proportional to the number of electron-hole pairs in the QD, i.e., γ N = N · γ [7]. As the excitation power density is increased and the bunching induced by a higher QD mean occupation number is strengthened, this approximation may become more questionable. has been determined as an average over various binning conditions.

S5. ESTIMATION OF THE DEFECT DENSITY THROUGH QUANTUM DOT EMISSION LINEWIDTH STATISTICS
As shown in Ref. [7], spectral diffusion in III-nitride polar QDs is also responsible for a very specific excitonic linewidth distribution, which is highlighted in Fig. 1d of the main text.
As detailed beforehand, QDs are impacted by the charging and discharging of neighboring DSs. While these charge fluctuations contribute to the broadening of exciton emission lines at cryogenic temperature, way beyond the natural linewidth, this feature can also be used to estimate the defect density in the dot surrounding.
To this end, we relied on the probability distribution function of the exciton linewidth for a given emission energy (E) that can be approached by a Fréchet distribution [7]: where the scaling parameter s and the shape parameter α are considered as free parameters here. The maximum value of f F r (E) is associated to a linewidth value (∆) that can then be used to estimate the point defect densityρ using [7]: where p is the occupation probability of a defect, e is the elementary charge, and κ is a numerically estimated coefficient that depends on the QD emission energy. Using p = 1/2 we can estimate a minimum value forρ. Ideally, ∆ is determined experimentally for a By taking advantage of the quasi-resonant excitation scheme adopted in this work, we have been able to record g (2) (τ ) traces with limited background emission at cryogenic tem- perature. Yet, as temperature increases, the thermal broadening of QD emission lines leads to an increased contribution of biexciton PL to the correlated counts and a reduction in the single-photon purity that follows accordingly. Thanks to the large biexciton binding energy of QDs emitting around 4.5 eV, single-photon emission could still be observed at RT, as described in the main text. We showed additionally that the resulting increase in g (2) (0) can be accounted for by considering the biexciton luminescence as an uncorrelated background [13]:  of g (2) (0) up to a 50 % collection efficiency of the exciton PL intensity, with g (2) (0) = 0.29 for a bandpass of 30 meV. Using such a bandpass would in turn enable us to perform measurements at lower excitation power density, reducing even more the contribution of the biexciton line. We thus expect our results to be amenable to improvement. In the main text, we reported on the bi-exponential dynamic that governs the exciton recombination whatever the QD emission energy. Several scenarios were postulated to account for this behavior. One plausible explanation relies on a PL signal arising from the two distinct QD exciton energy bright states. In this framework, the fast decay time τ S accounts simultaneously for the recombination of the high energy bright state B 2 and its relaxation toward lower energy states, i.e., toward the low energy bright state B 1 and the exciton dark states (cf. Fig. 2(a) of the main text). Since the long decay time τ L remains constant with temperature (cf. Fig. 9 of the main text), it is hinting at the a priori negligible impact of phonon-mediated processes for the states exhibiting this recombination dynamic. In other words, τ L is assumed to be mainly driven by the radiative recombination of B 1 . To account for this temperature-independence of τ L , a quasi-equilibrium should take place between B 1 and the dark states that occurs on a much faster timescale than τ L .
On the other hand, the weight of the fast component strongly depends on the refilling of B 2 , which occurs through absorption of phonons with an energy matching the splitting between B 2 and lower energy states. Hence, the enhancement of the PL intensity originating from the high energy bright state can occur either through an increase in temperature or a reduction in the bright state energy splitting. The latter coincides with an increase in QD size and can be monitored through energy-dependent TRPL measurements.
In order to test these predictions, we extracted the long-lived PL component of the TRPL transients by integrating the associated long-lived mono-exponential fit over the whole PL intensity decay profiles, (as shown in Fig. 8 of the main text). In each case, this component was normalized to the PL intensity decay profile integrated over the whole raw data (for delays τ > 0), hence leading to the intensity ratio denoted by the letter r in the main text.
When reaching low excitation power densities, the contribution from multi-excitonic states to the TRPL transients is expected to vanish so that r should saturate (r(τ → 0) → r 0 ). This is shown in Fig. S10 for QDs emitting at 3.80 eV. In the current framework, r 0 can be seen as a marker of the B 1 PL weight with respect to that of B 2 , as shown by Eq. (6) in the main text. In other words, the weight of B 1 should increase together with r 0 .
By monitoring a decrease in r 0 with decreasing QD emission energies (Fig. 10 of the main text), we showed that the long-lived decay contributes less to the PL of large QDs. This is in line with the expected decrease in the fine structure splitting with increasing QD size. From the ratios reported in Fig. S10, it appears now that r 0 is also decreasing with temperature for QDs emitting at 3.8 eV, as expected from the above-mentioned picture.
The variation remains, however, relatively small (reduction of about 30 %) and additional results are required to confirm the global trend. Similar measurements led on high energy QDs would prove extremely valuable. Indeed, the large fine structure splitting characteristic of small QDs results in a strong discrepancy between B 1 and B 2 recombination rates. At 5 K, radiative recombination from the high energy bright state (X 2 line) is thus thermally suppressed and the X 1 prevails. On the other hand, X 2 dominates PL spectra as soon as B 2 refilling from lower energy states becomes thermally activated (cf. Fig. S5). From TRPL data, this should translate into a large (r 0 → 1) and a small (r 0 → 0) r 0 value at 5 K and 300 K, respectively. Given the very low luminescence signal issued from GaN/AlN QD  Fig. S11). The latter ratio represents a substantial decrease from the r = 0.82 value obtained at 5 K under equivalent pumping conditions. Besides, r is observed to saturate faster at high temperature, as shown for QDs emitting near 3.80 eV (Fig. S10). Hence, the ratio r measured at 300 K for any excitation power density should be closer to the r 0 limit than its 5 K counterpart, i.e., r r 0 (P exc , 300 K) > r r 0 (P exc , 5 K) ∀ P exc . At 5 K and for QDs emitting at 4.25 eV, we determined r r 0 (0.35 W cm −2 ) > 0.9. We can thus reasonably expect the ratio r = 0.25 measured at RT under identical excitation power density (P exc = 0.35 W cm −2 ) to differ by less than 10 % from r 0 . As such, r 0 (300 K) < 0.3 for QDs emitting at 4.25 eV, which scales way below the r 0 (5 K) = 0.9 determined for similar QDs at cryogenic temperature.
This confirms that the short-lived TRPL component is thermally activated.

S9. CATHODOLUMINESCENCE OF THE UNETCHED GAN/ALN QUANTUM DOT SAMPLE
A representative cathodoluminescence (CL) emission spectrum recorded at T = 12 K using an acceleration voltage of 6 kV over an unetched area of the QD sample is shown in